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Let $F=\langle a,b\rangle$ be a free group of rank $2$. Its commutator subgroup has a nice free-basis: $$[a^m,b^n], \,\,\,\,m,n\in\mathbb{Z}.$$

Instead of $[F,F]$, we consider another simplest normal subgroup. Let $N$ be the smallest normal subgroup of $F$ containing $a$. Of course, $N$ will not only contain powers of $a$ but elements $bab^{-1}$ also, and in general $waw^{-1}$ for any word $w$ in $F$.

Question: What is a free-basis of $N$?

I was thinking analogously that $\{b^iab^{-i} \colon i\in\mathbb{Z}\}$ would be a free basis. But, this is not, since in the product of these elements, the end points will be $b$ or $b^{-1}$; in particular, $(ab)a(ab)^{-1}$ can not be obtained from this set.

[As noticed by Guerin, in last paragraph, after "But, this ...." is incorrect!]

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  • $\begingroup$ You have that $(ab)a(ab)^{-1}=b^0ab^{-0}bab^{-1}b^0a^{-1}b^{-0}$ so your element is in the set generated by your hypothetic base. $\endgroup$ – Clément Guérin Sep 24 '15 at 6:46
  • $\begingroup$ Oh! Thanks for pointing this. Then the mentioned set would be a generating set. Is it so? and is it "basis"? $\endgroup$ – Groups Sep 24 '15 at 6:48
  • $\begingroup$ Generating set property shouldn't be difficult, it suffices to show that for any $w$, $waw^{-1}$ can be written as a word in $b^iab^{-i}$ . I think this is easily handled by induction on the length of the word $w$. Free basis seems like a much harder point. $\endgroup$ – Clément Guérin Sep 24 '15 at 6:59
  • $\begingroup$ OK. I will try to prove at least that it is a generating set. $\endgroup$ – Groups Sep 24 '15 at 7:00
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For the free basis property. One way to state this is that any word $w$ written as a reduced non-trivial word $w_0$ of $e_i:=b^iab^{-i}$'s is non trivial. Write your word $w_0(e_1,....)=e_{f(1)}^{\epsilon(1)}...e_{f(r)}^{\epsilon(r)}$.

I claim that if the length of $w_0(e_1,...)$ is (as a word in $a$'s and $b$'s) $l_0$ then provided that $f=b^ka^{\epsilon}b^{-k}$ is different from $e_{f(r)}^{-\epsilon(r)}$ then the length of $w_0(e_1,...)f$ is strictly greater than the length $l_0$.

Use this to show that if $length(w_0(e_1,...))=0$ (as a word in $a$'s and $b$'s) then $w_0(e_1,...)$ is a trivial word (up to reduction) in $e_i$'s.

What you have shown then is that the surjective (because of the generating set thing we discussed) morphism from $\mathbb{F}_{\mathbb{Z}}$ to $N$ sending $e_i$ to $b^iab^{-i}$ is one to one. Whence $\{b^iab^{-i}\}$ is a free basis.

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